Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.10Divide Mixed Numbers

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can explain what dividing with mixed numbers means, rewrite them as improper fractions, find the quotient with common denominators or by multiplying by the reciprocal, and check it by multiplying.

Why This Matters Agent Chen is mapping a suspect's route. The total distance is 3 1/2 miles, and each segment between landmarks is 1/4 mile. She needs to know how many segments are on the route to place surveillance cameras at every landmark.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

WordWhat it meansExample
Mixed NumberSpanish: Número mixto A whole number plus a fraction, like 2 1/3. 2 1/3 means 2 wholes and 1/3 more — picture 2 full circles and 1/3 of another
Improper FractionSpanish: Fracción impropia A fraction where the top is bigger than or equal to the bottom, like 7/4. 7/3 is improper because 7 > 3. It equals 2 1/3 as a mixed number.
ConvertSpanish: Convertir To change a number to a new form but keep the same value. 2 1/3 → multiply 2 × 3 = 6, add 1 → 7/3. Same value, different form.
SimplifySpanish: Simplificar To write a fraction with smaller numbers that names the same amount, like 2/4 = 1/2. 8/6: GCF is 2 → 8÷2 / 6÷2 = 4/3 = 1 1/3
ReciprocalSpanish: Recíproco A fraction turned upside down. Dividing by a fraction gives the same answer as multiplying by its reciprocal. The reciprocal of 3/4 is 4/3. To divide by 3/4, multiply by 4/3.

How it worksWorked example

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Agent Chen has a 3 1/2-mile route cut into 1/4-mile segments. I want 3 1/2 ÷ 1/4.

  1. First I convert 3 1/2. Multiply the whole by the bottom: 3 × 2 = 6. Add the top: 6 + 1 = 7. So 3 1/2 = 7/2.
  2. Now I ask how many 1/4s fit in 7/2. In fourths, 7/2 = 14/4, and 14 fourths ÷ 1 fourth = 14 — the common-denominator method.
  3. The reciprocal shortcut gives the same answer: 7/2 × 4/1 = 28/2 = 14.
  4. Check by multiplying: 14 × 1/4 = 14/4 = 3 1/2. So 3 1/2 ÷ 1/4 = 14 segments.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Let's try 2 1/2 ÷ 1/2. First convert 2 1/2: 2 × 2 = 4, plus 1 = 5, so 2 1/2 = 5/2.

Answer:

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

  • Before dividing, I convert 2 1/2 to because it is easier to with improper fractions.

Word bank Mixed NumberImproper FractionConvertSimplifyReciprocalimproper5/2convertmultiplyreciprocalThe reciprocal shortcut

Watch out!A common mistake

A common mistake in Divide Mixed Numbers is dividing the whole-number parts and fraction parts separately instead of converting first

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.10Divide Mixed Numbers

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can explain what dividing with mixed numbers means, rewrite them as improper fractions, find the quotient with common denominators or by multiplying by the reciprocal, and check it by multiplying.

Why This Matters Agent Chen is mapping a suspect's route. The total distance is 3 1/2 miles, and each segment between landmarks is 1/4 mile. She needs to know how many segments are on the route to place surveillance cameras at every landmark.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Convert each mixed number to an improper fraction.

    Mixed NumberImproper Fraction
    1 3/4
    2 2/3
    4 1/2
    3 1/5

    Hint Every conversion keeps its denominator — 1 3/4 stays in fourths.

    Show your work
Part 2Practice the skillshow every step
  1. 2Circle the letter of the best answer. Show how you know.

    What is 1 1/2 ÷ 3/4?

    1. A1 1/8
    2. B2
    3. C3/4
    4. D1/2

    Hint 1 1/2 is a mixed number — it has to become one improper fraction before you divide.

    Show your work
  2. 3Circle the letter of the best answer. Show how you know.

    What is 2 1/4 ÷ 3/4?

    1. A1 1/2
    2. B2 1/3
    3. C3
    4. D9/16

    Hint Convert 2 1/4 to an improper fraction before anything else.

    Show your work
  3. 4Circle the letter of the best answer. Show how you know.

    What is 3 1/2 ÷ 1/4?

    1. A3 1/8
    2. B7/8
    3. C13/4
    4. D14

    Hint Two steps: convert 3 1/2 first, then divide by 1/4.

    Show your work
Part 3Apply itreal-world problems
  1. 5Circle the letter of the best answer. Show how you know.

    Convert 3 2/5 to an improper fraction.

    1. A6/5
    2. B15/5
    3. C17/5
    4. D32/5

    Hint Converting keeps the denominator 5 the same — only the top number changes.

    Rewrite each fraction
    Work it out
    Simplify
    1. 1Same size piecesCommon denominator first.
    2. 2OperateOnly then add or subtract.
    3. 3SimplifyDivide out common factors.
Part 4Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1Problem1 1/2 ÷ 3/4
    2. 2Convert1 1/2 = 3/2
    3. 3Multiply by the reciprocal3/2 × 3/4
    4. 4Multiply9/8 = 1 1/8

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Hint Look at the reciprocal step — did the divisor get flipped?

    Correct work
    Correct answer:

Explain your thinkingWhy is converting to an improper fraction the first step when dividing mixed numbers?

Sentence starter Before dividing, I convert 2 1/2 to ___ because it is easier to ___ with improper fractions.

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.10Divide Mixed Numbers

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can explain what dividing with mixed numbers means, rewrite them as improper fractions, find the quotient with common denominators or by multiplying by the reciprocal, and check it by multiplying.

Why This Matters Agent Chen is mapping a suspect's route. The total distance is 3 1/2 miles, and each segment between landmarks is 1/4 mile. She needs to know how many segments are on the route to place surveillance cameras at every landmark.

Part 1Understand the ideamodels and meaning
  1. 1Complete the table. Show how you found each value.

    Convert each mixed number to an improper fraction, then divide.

    ProblemConvert to ImproperThe reciprocal shortcutQuotient
    3 1/2 ÷ 1/4
    1 3/4 ÷ 1/2
    2 1/3 ÷ 7/6
    4 1/5 ÷ 3/5
    Show your work
  2. 2Write the letter of the matching item on each line.

    Match each mixed number division to its quotient.

    1. 2 1/4 ÷ 3/4
    2. 4 1/2 ÷ 9/8
    3. 1 2/3 ÷ 5/6
    4. 3 1/3 ÷ 2/3
    5. 5 1/4 ÷ 3/4
    6. 1 1/2 ÷ 1/4
    • A3
    • B4
    • C2
    • D5
    • E7
    • F6
  3. 3Use the model to solve. Show your work.
    2 1/2 = 5/2 = five halves1/21/21/21/21/2

    How many 1/2-size pieces fit into 2 1/2?

    Show your work
    Answer:
Part 2Apply itreal-world problems
  1. 4Circle the letter of the best answer. Show how you know.

    A baker has 2 1/4 cups of butter. Each batch of cookies needs 3/4 cup of butter. How many batches can the baker make?

    1. A1 1/2 batches
    2. B2 batches
    3. C3 batches
    4. D6 batches
    Show your work
  2. 5Circle the letter of the best answer. Show how you know.

    Agent Chen has walked 2 1/4 miles, which is 3/4 of her whole route. How long is the whole route? (2 1/4 ÷ 3/4 asks how much is in one whole route.)

    1. A1 11/16 miles
    2. B3 miles
    3. C1/3 mile
    4. D2 1/3 miles
    Show your work
Part 3Explain and correctreasoning
  1. 6Find the mistake. Explain it, then show the correct work.

    Spot the Common Mistake

    1. 1Problem3 1/3 ÷ 2/3
    2. 2Multiply by the reciprocal (skipped converting)3 1/3 × 3/2
    3. 3Multiply the fraction part only1/3 × 3/2 = 3/6 = 1/2
    4. 4Combine with the whole3 + 1/2 = 3 1/2

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingWhy is converting to an improper fraction the first step when dividing mixed numbers?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help
Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.NOS.1

Lesson 6.10Divide Mixed Numbers

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice
Name Date Period

Learning target I can explain what dividing with mixed numbers means, rewrite them as improper fractions, find the quotient with common denominators or by multiplying by the reciprocal, and check it by multiplying.

Why This Matters Agent Chen is mapping a suspect's route. The total distance is 3 1/2 miles, and each segment between landmarks is 1/4 mile. She needs to know how many segments are on the route to place surveillance cameras at every landmark.

  1. 1Find the mistake. Explain it, then show the correct work.

    Find the Mixed Number Error

    1. 1Problem2 1/3 ÷ 1/2
    2. 2Convert2 1/3 = 6/3
    3. 3Multiply by the reciprocal6/3 × 2/1
    4. 4Multiply12/3 = 4

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:
  2. 2Answer in complete sentences.

    A detective has a rope that is 5 1/4 feet long. She needs to cut it into pieces that are each 3/4 foot for tying evidence tags. How many pieces can she cut? Write the equation, show each step (convert, the reciprocal shortcut, multiply, simplify), and verify your answer.

    Write your answer
  3. 3Find the mistake. Explain it, then show the correct work.

    Find Liam's Mistake

    1. 1Problem3 1/2 ÷ 1 1/4
    2. 2Convert first number3 1/2 = 7/2
    3. 3Convert second number1 1/4 = 5/4
    4. 4Divide7/2 ÷ 5/4 = 7/2 × 5/4 = 35/8 = 4 3/8

    Which step has the mistake? Explain what went wrong, then show the correct work.

    Correct work
    Correct answer:

Explain your thinkingWhy is converting to an improper fraction the first step when dividing mixed numbers?

3-Point Exit Checklist: ☐ 1. Units & Labels Verified ☐ 2. Mathematical Reasoning Stated ☐ 3. Calculations Double-Checked
How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help